That's basically what you would do thousands of times when you build a BSP tree.
First of all, don't think in terms of the x, y or z axis. If you split a 3d object you need a plane. This plane could be defined by a normal and a distance along the normal from origo.
With triangles completely on either side, it's trivial. But some triangles will probably intersect the split-plane. They need to be clipped.
Google will serve you from here...